Lower bounds for resonances of infinite-area Riemann surfaces

Abstract

For infinite area, geometrically finite surfaces X = Γ\H, we prove new lower bounds on the local density of resonances D(z) when z lies in a logarithmic neighborhood of the real axis. These lower bounds involve the dimension δ of the limit set of Γ. The first bound is valid when δ > 1 2 and shows logarithmic growth of the number D(z) of resonances at high energy i.e. when |Re(z)| → +∞. The second bound holds for δ > 3 4 and if Γ is an infinite index subgroup of certain arithmetic groups. In this case we obtain a polynomial lower bound. Both results are in favor of a conjecture of Guillope-Zworski on the existence of a fractal Weyl law for resonances.